I. Field of the Invention
The present invention relates to cellular communications and also relates to the Nyquist rate for data symbol transmission, the Shannon bound on communications capacity, and symbol modulation and demodulation for high-data-rate satellite, airborne, wired, wireless, and optical communications and includes all of the communications symbol modulations and the future modulations for single links and multiple access links which include electrical and optical wired, mobile, point-to-point, point-to-multipoint, multipoint-to-multipoint, cellular, multiple-input multiple-output MIMO, and satellite communication networks. In particular it relates to WiFi, WiMax and long-term evolution LTE for cellular communications and satellite communications. WiFi, WiMax use orthogonal frequency division multiplexing OFDM on both links and LTE uses SC-OFDM on the uplink from user to base station and OFDM on the downlink form base station to user. WiMax occupies a larger frequency band than WiFi and both use OFDM waveforms. SC-OFDM is a single carrier orthogonal waveform version of OFDM which uses orthogonal frequency subbands of varying widths.
II. Description of the Related Art
Two fundamental bounds on communications are the Nyquist rate and the Shannon capacity theorem. The Nyquist rate is the complex digital sampling rate equal to B that is sufficient to include all of the information within a frequency band B. For communications, equivalent expressions for the Nyquist rate bound are defined in equations (1).Ts≧1/B  (1)BTS≧1wherein 1/Ts is the data symbol transmission rate in the frequency band B which means Ts is the spacing between the data symbols.
The Shannon bound for the maximum data rate C is a bound on the corresponding number of information bits per symbol b as well as a bound on the communications efficiency η and is complemented by the Shannon coding theorem, and are defined in equations (2).
                                                                                          Shannon  bounds  and  coding  theorem                                ⁢                                                                  ⁢                                  1  Shannon  capacity  theorem                                            ⁢                                                          ⁢                                                                                          C                      =                                            ⁢                                                                        B                          ⁢                                                                                                          ⁢                                                                                    log                              2                                                        ⁡                                                          (                                                              1                                +                                                                  S                                  ⁢                                                                      /                                                                    ⁢                                  N                                                                                            )                                                                                ⁢                                                                                                          ⁢                                                      Channel  capacity  in  bit/second                                                                          =                        Bps                                                                                                                                                                              ⁢                                                                        for  an  additive   white  Gaussian  noise                                                ⁢                                                                                                  ⁢                        AWGN                        ⁢                                                    channel  with  bandwidth                                                                                                                                                                                                      ⁢                                              B                        ⁢                                                                                                  ⁢                                                  wherein                                                   ⁢                                                  “                                                      log                            2                                                    ”                                                ⁢                                                                                                  ⁢                                                  is  the  logarithm  to  the base  2                                                                                                                                                                                =                                            ⁢                                              Maximum   rate  at  which  information  can  be reliably   transmitted   over                                                                                                                                                                              ⁢                                              a  noisy  channel  where  S/N  is  the signal-to-noise  ratio  in  B                                                                                                        ⁢                                                          ⁢                                                                    2  Shannon  bound  on  b,                                    ⁢                                                                          ⁢                  η                                ,                                                                  ⁢                                                      and                                    ⁢                                                                          ⁢                                      E                    b                                    ⁢                                      /                                    ⁢                                      N                    o                                                              ⁢                                                          ⁢                                                                                                                  max                        ⁢                                                  {                          b                          }                                                                    =                                            ⁢                                              max                        ⁢                                                  {                                                      C                            ⁢                                                          /                                                        ⁢                            B                                                    }                                                                                                                                                                                =                                            ⁢                                                                        log                          2                                                ⁡                                                  (                                                      1                            +                                                          S                              ⁢                                                              /                                                            ⁢                              N                                                                                )                                                                                                                                                                                =                                            ⁢                                              max                        ⁢                                                  {                          η                          }                                                                                                                                                                                                                                  E                          b                                                ⁢                                                  /                                                ⁢                                                  N                          o                                                                    =                                            ⁢                                                                                                    [                                                                                                                            2                                  ^                                  max                                                                ⁢                                                                  {                                  b                                  }                                                                                            -                              1                                                        ]                                                    /                          max                                                ⁢                                                  {                          b                          }                                                                                                                                                                                                                                  wherein                                                  ⁢                        b                                            =                                            ⁢                                                                        C                          ⁢                                                      /                                                    ⁢                          B                          ⁢                                                                                                          ⁢                          in                          ⁢                                                                                                          ⁢                          Bps                          ⁢                                                      /                                                    ⁢                          Hz                                                =                                                  Bits                          ⁢                                                      /                                                    ⁢                                                      symbol                                                                                                                                                                                                                                  η                        =                                                ⁢                                                  b                          ⁢                                                      /                                                    ⁢                                                      T                            s                                                    ⁢                          B                                                                    ,                                                                                          ⁢                                              Bps                        ⁢                                                  /                                                ⁢                        Hz                                                                                                                                                                                T                        s                                            =                                            ⁢                                              symbol  interval                                                                                                                                                    (              2              )                                                                      
3 Shannon coding theorem for the information bit rate Rb                 For Rb<C there exists codes which support reliable communications        For Rb>C there are no codes which support reliable communicationswherein Eb/No is the ratio of energy per information bit Eb to the noise power density No, max{b} is the maximum value of the number of information bits per symbol b and also is the information rate in Bps/Hz, and since the communications efficiency η=b/(TSB) in bits/sec/Hz it follows that maximum values of b and η are equal. Derivation of the equation for Eb/No uses the definition Eb/No=(S/N)/b in addition to 1 and 2. Reliable communications in the statement of the Shannon coding theorem 3 means an arbitrarily low bit error rate BER.        
OFDM is defined in FIG. 1 for the WiFi 802.16 standard power spectrum in 1,2 which implements the inverse FFT (IFFT=FFT−1) to generate OFDM (or equivalently OFDMA which is orthogonal frequency division multiple access to emphasize the multiple access applications) data symbol tones 2 over the first 3.2 μs of the 4 μS data packet in 30 in FIG. 7 with some rolloff of the tones at their ends for spectral containment. Data symbol tones are modulated with 4 PSK, 16QAM, 64QAM, 256QAM depending on the transmission range and data rate and for 256QAM using the code rate option R=¾ yields the information rate b=6 Bps/Hz for the WiFi standard, with other code options available. The N=64 point FFT−1 generates N=64 tones in 2 over the 20 MHz WiFi band with 48 tones used for data transmission. In 3 the WiFi parameters are defined including a calculation of the maximum data rate Rb=57 Mbps. Later versions of WiFi allow WiFi bands of 1.25, 5, 10, 20 MHz corresponding to N=4, 16, 32, 64. For this representative OFDM WiFi QLM disclosure we are considering the WiFi standard in FIG. 1. The maximum data rate supported by WiFi standard is calculated in 3 to be ˜57 Mbps using 256QAM modulation and wherein “˜” represents an approximate value. OFDM uses pulse waveforms in time and relies on the OFDM tone modulation to provide orthogonality. SC-OFDM is a pulse-shaped OFDM that uses shaped waveforms in time to roll-off the spectrum of the waveform between adjacent channels to provide orthogonality, allows the user to occupy subbands of differing widths, and uses a different tone spacing, data packet length, and sub-frame length compared to OFDM for WiFi, WiMax.